Optimal. Leaf size=31 \[ \frac{e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac{e^{3 x}}{3} \]
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Rubi [A] time = 0.019334, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 270} \[ \frac{e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac{e^{3 x}}{3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 270
Rubi steps
\begin{align*} \int \left (-e^{-x}+e^x\right )^3 \, dx &=\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3}{x^4} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-3-\frac{1}{x^4}+\frac{3}{x^2}+x^2\right ) \, dx,x,e^x\right )\\ &=\frac{e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac{e^{3 x}}{3}\\ \end{align*}
Mathematica [A] time = 0.0131351, size = 30, normalized size = 0.97 \[ \frac{1}{3} e^{-3 x} \left (-9 e^{2 x}-9 e^{4 x}+e^{6 x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 24, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{3}}{3}}-3\,{{\rm e}^{x}}+{\frac{1}{3\, \left ({{\rm e}^{x}} \right ) ^{3}}}-3\, \left ({{\rm e}^{x}} \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.922947, size = 31, normalized size = 1. \begin{align*} \frac{1}{3} \, e^{\left (3 \, x\right )} - 3 \, e^{\left (-x\right )} + \frac{1}{3} \, e^{\left (-3 \, x\right )} - 3 \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73138, size = 70, normalized size = 2.26 \begin{align*} \frac{1}{3} \,{\left (e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.126128, size = 24, normalized size = 0.77 \begin{align*} \frac{e^{3 x}}{3} - 3 e^{x} - 3 e^{- x} + \frac{e^{- 3 x}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06519, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{3} \,{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{3} \, e^{\left (3 \, x\right )} - 3 \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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